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A190014 a(0)=0, a(1)=1, if n = (n-1)', then a(n)=0 otherwise a(n)=2*a((n-1)'), where n' is the arithmetic derivative of n.

%I A190014 #13 Apr 10 2025 08:12:42
%S A190014 0,1,0,2,2,4,2,8,2,4,4,16,2,8,2,8,4,4,2,8,2,4,8,16,2,4,8,16,32,4,2,0,
%T A190014 2,4,4,16,4,4,2,8,8,4,2,8,2,4,16,8,2,32,4,8,4,16,2,512,8,8,16,0,2,8,2,
%U A190014 8,16,4,4,16,2,4,16,0,2,16,2,16,1024,4,4,0,2,256,4,16,2,8,16,8,4,4,2,32,4,8,8,64,4,4,2,8,32,4
%N A190014 a(0)=0, a(1)=1, if n = (n-1)', then a(n)=0 otherwise a(n)=2*a((n-1)'), where n' is the arithmetic derivative of n.
%C A190014 Only power of 2 and zeros. If p is prime than a(p+1)=2.
%C A190014 If n’>n+1 than a(n+1) is not immediately available. It is necessary to find a(n’)=2*a((n’-1)’) and, if necessary, to repeat the process until a term can be calculate. For instance:
%C A190014 a(9)=2*a(12) -> a(12)=2 and therefore a(9)=4.
%C A190014 Again: a(55)=2*a(81) -> a(81)=2*a(176) -> a(176)=8*a(112) -> a(112)=16 and therefore a(176)=128 -> a(81)=256 -> a(55)=512.
%C A190014 First zero at a(31)=2*a(31) and for all Giuga numbers plus one (31, 859, 1723, 66199, etc.). This because the so far known Giuga numbers satisfy the equation n’=n+1. Other zeros for a(59)=8*a(31), a(71)=16*a(31), a(79)=32*a(31),  a(106)=32*a(31), etc.
%C A190014 The general equation a(n+1)=k*a(n’), with k integer and |k|>1, a(0)=0, a(1)=1, leads to the following sequence:  0, 1, 0, k, k, k^2, k, k^3,k, k^2, k^2, k^4, k, k^3, k, k^3, k^2, k^2, k, k^3, etc.
%C A190014 For k=1 or k=-1 we get and incongruence because of a(31)=a(31).
%H A190014 Paolo P. Lava, <a href="/A190014/b190014.txt">Table of n, a(n) for n = 0..5000</a>
%H A190014 Paolo P. Lava, <a href="/A190014/a190014.pdf">Plot of the first 5000 terms of the sequence</a>
%e A190014 a(0)=0
%e A190014 a(1)=1
%e A190014 a(2)=a(1+1)=2*a(1')=2*a(0)=0
%e A190014 a(3)=a(2+1)=2*a(2')=2*a(1)=2
%e A190014 a(4)=a(3+1)=2*a(3')=2*a(1)=2
%e A190014 a(5)=a(4+1)=2*a(4')=2*a(4)=4
%e A190014 a(6)=a(5+1)=2*a(5')=2*a(1)=2
%e A190014 a(7)=a(6+1)=2*a(6')=2*a(5)=8  etc.
%p A190014 with(numtheory);
%p A190014 P:=proc(i)
%p A190014 local a,f,n,p,pfs,t;
%p A190014 a:=array(0..100000); a[0]:=0; a[1]:=1; t:=2; lprint(0,a[0]); lprint(1,a[1]);
%p A190014 for n from 1 by 1 to i do
%p A190014     pfs:=ifactors(n)[2]; f:=n*add(op(2,p)/op(1,p),p=pfs);
%p A190014     a[n+1]:=t*a[f]; lprint(n+1,a[n+1]);
%p A190014 od;
%p A190014 end:
%t A190014 dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Module[{d = dn[n - 1]}, If[d == n, 0, 2 a[d]]]; Array[a,100,0] (* _T. D. Noe_, May 05 2011 *)
%Y A190014 Cf. A003415, A007850,
%K A190014 nonn
%O A190014 0,4
%A A190014 _Paolo P. Lava_, May 04 2011